# Uncertainty characterisation

## Variability and incertitude

Modern risk analysts distinguish between variability and incertitude. ***Variability*** (also called randomness, aleatory uncertainty, or irreducible uncertainty) arises from natural stochasticity, environmental or structural variation across space or time, manufacturing heterogeneity among components or individuals. ***Incertitude***, also called ignorance, epistemic uncertainty, subjective uncertainty or reducible uncertainty, arises from incompleteness of knowledge. Sources of incertitude include measurement uncertainty, small sample sizes, and data censoring, ignorance about the details of physical mechanisms and processes.

For an engineering analysis, the **challenge** lies in formulating suitable uncertainty models given available information, **without introducing unwarranted assumptions**. However, the available information is often vague, ambiguous, or qualitative. Available data are frequently limited and of poor quality, giving rise to challenges in eliciting precise probabilistic specifications. Solutions to this problem are discussed in the literature, under the framework of imprecise probability, from various perspectives using different mathematical concepts, including for example random sets, evidence theory, fuzzy stochastic concepts, info-gap theory, and probability bounds analysis.

```{tip}
It is suggested to use interval analysis for propagating ignorance and the methods of probability theory for propagating variability.
```



## Bounding distributional parameters

The mean of a normal distribution may be elicited from an expert, but this expert cannot be precise to a certain value but rather give a range based on past experience.

````{tab} verbose
To comprehensively characterise a pbox, specify the bounds for the parameters along with many other ancillary fields.

```python
from pyuncertainnumber import UncertainNumber as UN

e = UN(
    name='elas_modulus', 
    symbol='E', 
    units='Pa', 
    essence='pbox', 
    distribution_parameters=['gaussian', ([0,12],[1,4])])
```
````

````{tab} shortcut
In cases where one wants to do computations quickly.

```python
import pyuncertainnumber as pun
un = pun.norm([0,12],[1,4])
```
````


````{tab} pba API
For low-level controls and customisation

```python
from pyuncertainnumber import pba
pbox = pba.normal([0,12],[1,4])
```
````


```{tip}
The different sub-types of uncertain number can normally convert to one another (though may not be one by one), ergo the uncertain number been said to be a unified representation.
```

```{seealso}
See also the tutorial the [What is an uncertain number](https://pyuncertainnumber.readthedocs.io/en/latest/tutorials/what_is_un.html) to get started.
```


## Aggregation of multiple sources of information

Expert elicitation has been a challenging topic, especially when knowledge is limited and measurements are sparse. Multiple experts may not necessarily agree on the choice of elicited probability distributions, which leads to the need for aggregation. Below shows two situations for illustration.

Assume the expert opinions are expressed in closed intervals. There may well be multiple such intervals from different experts and these collections of intervals can be overlapping, partially contradictory or even completely contradictory. Their relative credibility may be expressed in probabilities. Essentially such information creates a **Dempster-Shafer structure**. On the basis of a mixture operation, such information can be aggregated into a **p-box**.

```{seealso}
See also the tutorial [uncertainty aggregation](https://pyuncertainnumber.readthedocs.io/en/latest/tutorials/uncertainty_aggregation.html) to get started.
```

## Inter-variable dependence

P-box arithmetic also extends the convolution of probability distributions which has typically been done with the independence assumption. However, often in engineering modelling practices independence is assumed for mathematical easiness rather than warranted. Fortunately, the uncertainty about the dependency between random variables can be characterised by the probability bounds, as seen below. It should be noted that such dependency bound does not imply independence.

```{image} ../../../assets/addition_bound.png
:alt: sum of two random variables without dependency specification
:class: bg-primary
:width: 400px
:align: center

The sum of two random variables of lognormal distribution without dependency specification
```
```{seealso}
See also the tutorial [depenency structure](https://pyuncertainnumber.readthedocs.io/en/latest/examples/characterisation/example_dependency_dev_purpose.html) to get started .
```



## Known statistical properties

When the knowledge of a quantity is limited to the point where only some statistical information is available, such as the *min*, *max*, *median* etc. but not about the distribution and parameters, such partial information can serve as **constraints** to bound the underlying distribution:

```{image} ../../../assets/known_constraints.png
:alt: known constraints
:class: bg-primary
:width: 1000px
:align: center
```

```{seealso}
See also the tutorial [characterise as you go](https://pyuncertainnumber.readthedocs.io/en/latest/examples/characterisation/characterise_what_you_know.html) to get started.
```


## Hedged numerical expression

Sometimes only purely qualitative information is available. An important part of processing elicited numerical inputs is an ability to quantitatively decode natural-language words, the linguistic information, that are commonly used to express or modify numerical values. Some examples include ‘about’, ‘around’, ‘almost’, ‘exactly’, ‘nearly’, ‘below’, ‘at least’, ‘order of’, etc. A numerical expression with these approximators are called *hedges*. Extending upon the significant-digit convention, a series of interval interpretations of common hedged numerical expressions are proposed.

```{image} ../../../assets/interval_hedge.png
:alt: interval hedges
:class: bg-primary
:width: 1000px
:align: center

Symmetric (left) and asymmetric (right) approximators of the number 7
```

Besides intervals, `PyUncertainNumber` also supports interpreting hedged expressions into p-boxes. As an example, assume one wants to find out what "about" is about in terms of the uncertainty. The syntax and result is shown below:

```python
import pyuncertainnumber as pun
pun.hedge_interpret('about 200', return_type='pbox').display()
```

```{image} ../../../assets/about_200.png
:alt: about 200
:class: bg-primary
:width: 400px
:align: center

hedged numerical expression "about 200"
```

```{seealso}
See also the tutorial [Interpret linguistic hedges](https://pyuncertainnumber.readthedocs.io/en/latest/examples/characterisation/linguistic_approximation.html) to get started.
```


## Data uncertainty

Measurement uncertainty is another main source uncertainty of data uncertainty besides sampling uncertainty. Point estimates from samples vary from one to another. We will typically use confidence intervals (as interval estimators) to account for the sampling uncertainty. As an example, `PyUncertainNumber` provides support for Kolmogorov–Smirnov (KS) confidence limits to infer the confidence limits for empirical cumulative distribution function.

```{seealso}
See also the [confidence box](../cbox.md) for a distributional estimator.
```

As to measurement uncertainty, `Intervals` turn out to be a natural means of mathematical construct for imprecise data due to the common understanding of margin of error, which leads to the midpoint notation of an interval object. `PyUncertainNumber` provides an extension of the Kolmogorov–Smirnov confidence limits for interval-valued data as well. The lower figure shows such confidence limits for the skinny data.

```{figure} ../../../assets/ks_precise.png
:alt: Kolmogorov–Smirnov bounds for precise data
:class: bg-primary
:align: center
:width: 400px
```

```{figure} ../../../assets/ks_imprecise.png
:alt: Kolmogorov–Smirnov bounds for imprecise data
:class: bg-primary
:align: center
:width: 400px
```